BEAM FORMATION WITH ELECTRON GUNS 393 



farther out. Since item (1) is influenced by item (2), the specific as- 

 sumptions involved in the latter case will be treated first. 



When current density is uniform across the beam and its cross section 

 changes slowly with distance, considerations of the type outlined above 

 for the gun region show that those thermal electrons which remain 

 within the beam will continue to have a Gaussian distribution with re- 

 spect to a non-thermal electron emitted from the same cathode point. 

 When current density is not uniform over the cross section, we would 

 still like to preserve the mathematical simplicity of obtaining the current 

 density as a function of beam radius merely by superposing Gaussian 

 distributions which can be associated with each non-thermal electron. 

 To lessen the error involved in this simplified approach, we will arrive 

 at a value for the standard deviation, a (which specifies the Gaussian 

 distribution), in a rather special way. In particular, a at any axial po- 

 sition, z, will be taken as the radial coordinate of an electron emitted 

 from the center of the cathode with a transverse velocity of emission 

 given by, 



ve = y- 



— (21) 



m 



It is clear from (17) that for such an electron, r = o- in the gun region. 

 From (18), the fraction of the electrons from a common point on the 

 cathode which will have r ^ a in the gun region is 



2 



fraction = [ e'^'-'"-''^ d ^= I - e'"' = 0.393 (22) 



If re denotes the radial position of the outermost non-thermal electron 

 and if 0- > /■,, , the "a--electron" will be moving in a region where the 

 space charge density is significantly lower than at the axis. We could, 

 of course, have followed the path of an electron with initial velocity 

 equal to say 0.1 or 10 times that given in (21) and called the correspond- 

 nig radius O.lcr or lOo-. The reason for preferring (21) is that about 0.4 

 or nearly half of the thermal electrons emitted from a common cathode 

 point will have wandered a distance less than a from the path of a non- 

 thermal electron emitted from the same cathode point, while other 

 thermal electrons will ha\'e wandered farther from this path; conse- 

 quently, the current density in the region of the o--electron is expected 

 to be a reasonable average on which beam spreading due to thermal 

 \elocities may be based. With this understanding of how a is to be cal- 

 culated, we can proceed to the calculation of non-thermal electron 

 trajectories as suggested in item (1). 



